keyword:walk - cp's OEIS frontend

A151541 Number of 2-sided triangular strip polyedges with n cells.

1, 3, 8, 32, 123, 523, 2201, 9443, 40341, 172649, 736926, 3141607, 13367012, 56790498, 240919918, 1020753475, 4319803799, 18262494912, 77134873774, 325518862387, 1372679840360, 5784417772262

Offset: 1

Ed Pegg Jr, May 13 2009

Also number of unrooted self-avoiding walks of n steps on hexagonal [ =triangular ] lattice. - Hugo Pfoertner, Jun 23 2018

Ed Pegg, Jr., Illustrations of polyforms
Hugo Pfoertner, Illustration of ratio A001334(n)/a(n) using Plot2.
Eric Weisstein's World of Mathematics, Polyedge

Cf. A037245, A151539, A151540, A159867, A266925.

Asymptotically approaches (1/24)*A001334(n) for increasing n.

a(9)-a(13) from Joseph Myers, Oct 05 2011

a(14)-a(22) from Bert Dobbelaere, Mar 23 2025

A152746 Six times hexagonal numbers: 6n(2*n-1).

Original entry on oeis.org

0, 6, 36, 90, 168, 270, 396, 546, 720, 918, 1140, 1386, 1656, 1950, 2268, 2610, 2976, 3366, 3780, 4218, 4680, 5166, 5676, 6210, 6768, 7350, 7956, 8586, 9240, 9918, 10620, 11346, 12096, 12870, 13668, 14490, 15336, 16206, 17100

Offset: 0

Omar E. Pol, Dec 12 2008

Sequence found by reading the line from 0, in the direction 0, 6, ..., in the square spiral whose vertices are the generalized octagonal numbers A001082. - Omar E. Pol, Sep 18 2011

a(n) is the number of walks on a cubic lattice of n dimensions that return to the origin, not necessarily for the first time, after 4 steps. - Shel Kaphan, Mar 20 2023

Ivan Panchenko, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (3,-3,1).

Cf. A000384, A001082, A002939, A094159, A085250, A152745.

Column n=2 of A287318.

[6*n*(2*n-1): n in [0..50]]; // G. C. Greubel, Sep 01 2018

6*PolygonalNumber[6,Range[0,40]] (* The program uses the PolygonalNumber function from Mathematica version 10 *) (* Harvey P. Dale, Mar 04 2016 *)
LinearRecurrence[{3,-3,1}, {0,6,36}, 50] (* or *) Table[6*n*(2*n-1), {n,0,50}] (* G. C. Greubel, Sep 01 2018 *)

a(n)=6*n*(2*n-1) \\ Charles R Greathouse IV, Jun 17 2017

a(n) = 12*n^2 - 6*n = A000384(n)*6 = A002939(n)*3 = A094159(n)*2.
a(n) = a(n-1) + 24*n - 18 (with a(0)=0). - Vincenzo Librandi, Nov 26 2010
From G. C. Greubel, Sep 01 2018: (Start)
G.f.: 6*x*(1+3*x)/(1-x)^3.
E.g.f.: 6*x*(1+2*x)*exp(x). (End)
From Amiram Eldar, Mar 30 2023: (Start)
Sum_{n>=1} 1/a(n) = log(2)/3.
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/12 - log(2)/6. (End)

A276852 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

Original entry on oeis.org

0, 1, 2, 7, 28, 121, 560, 2677, 13230, 66742, 343092, 1788681, 9439870, 50321865, 270594896, 1465941763, 7993664588, 43839212778, 241650560756, 1338084935826, 7439615051328, 41516113036777, 232452845782308, 1305500166481715, 7352433083806020, 41514430735834714

Offset: 0

Michael Wallner, Sep 21 2016

Alois P. Heinz, Table of n, a(n) for n = 0..1292
C. Banderier, C. Krattenthaler, A. Krinik, D. Kruchinin, V. Kruchinin, D. Nguyen, and M. Wallner, Explicit formulas for enumeration of lattice paths: basketball and the kernel method, arXiv preprint arXiv:1609.06473 [math.CO], 2016.

Cf. A276901, A276902, A276903, A276904.

walks[n_, k_, h_] = 0;
walks[1, k_, h_] := Boole[0 < k <= h];
walks[n_, k_, h_] /; n >= 2 && k > 0 := walks[n, k, h] = Sum[walks[n - 1, k - x, h], {x, h}] + Sum[walks[n - 1, k + x, h], {x, h}];
(* walks represents the number of positive walks with n steps {-h, -h+1, ... -1, 1, ..., h} that end at altitude k *)
A276852[n_] := (Do[walks[m, k, 3], {m, n}, {k, 3 m}]; walks[n, 1, 3]) (* Davin Park, Oct 10 2016 *)

A001184 Number of simple Hamiltonian paths connecting opposite corners of a 2n+1 X 2n+1 grid.

Original entry on oeis.org

1, 2, 104, 111712, 2688307514, 1445778936756068, 17337631013706758184626, 4628650743368437273677525554148, 27478778338807945303765092195103685118924

Offset: 0

Don Knuth, Dec 07 1995

Cf. A121788, A333580.

a(n) = A121788(2n), n>0. - Ashutosh Mehra, Dec 19 2008

a(7)-a(8) copied from A121788 by Alois P. Heinz, Sep 27 2014

A010575 Number of n-step self-avoiding walks on 4-d cubic lattice.

Original entry on oeis.org

1, 8, 56, 392, 2696, 18584, 127160, 871256, 5946200, 40613816, 276750536, 1886784200, 12843449288, 87456597656, 594876193016, 4047352264616, 27514497698984, 187083712725224, 1271271096363128, 8639846411760440, 58689235680164600, 398715967140863864

Offset: 0

N. J. A. Sloane

The computation for n=16 took 11.5 days CPU time on a 500MHz Digital Alphastation. The asymptotic behavior lim n->infinity a(n)/mu^n=const is discussed in the MathWorld link. The Pfoertner link provides an illustration of the asymptotic behavior indicating that the connective constant mu is in the range [6.79,6.80]. - Hugo Pfoertner, Dec 14 2002

Computation of the new term a(17) took 16.5 days CPU time on a 1.5GHz Intel Itanium 2 processor. - Hugo Pfoertner, Oct 19 2004

Hugo Pfoertner, Table of n, a(n) for n = 0..24 [from the Clisby et al. link below]
N. Clisby, R. Liang, and G. Slade, Self-avoiding walk enumeration via the lace expansion, J. Phys. A: Math. Theor., vol. 40 (2007), p. 10973-11017, Table A6 for n <= 24.
Nathan Clisby, Monte Carlo study of four-dimensional self-avoiding walks of up to one billion steps, arXiv:1703.10557 [cond-mat.stat-mech], 30 Mar 2017.
M. E. Fisher and D. S. Gaunt, Ising model and self-avoiding walks on hypercubical lattices and high density expansions, Phys. Rev. 133 (1964) A224-A239.
D. MacDonald, D. L. Hunter, K. Kelly, and N. Jan, Self-avoiding walks in two to five dimensions: exact enumerations and series study, J Phys A: Math Gen 25 (1992) Vol. 6, 1429-1440 [Gives 18 terms]
A. M. Nemirovsky et al., Marriage of exact enumeration and 1/d expansion methods: lattice model of dilute polymers, J. Statist. Phys., 67 (1992), 1083-1108.
Hugo Pfoertner, Results for the 4D Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk Connective Constant

Cf. A001411, A001412, A242355, A323856, A323857, A366925.

c A "brute force" Fortran program to count the 4D walks is available at the Pfoertner link.

a(n) = 8*A366925(n) for n >= 1. - Hugo Pfoertner, Nov 03 2023

a(12)-a(16) from Hugo Pfoertner, Dec 14 2002
a(17) from Hugo Pfoertner, Oct 19 2004
a(18) onwards from R. J. Mathar using data from Clisby et al, Aug 31 2007

A038373 Number of n-step self-avoiding paths on quadrant grid starting at quadrant origin.

Original entry on oeis.org

1, 2, 4, 10, 24, 60, 146, 366, 912, 2302, 5800, 14722, 37368, 95304, 243168, 622518, 1594622, 4094768, 10521384, 27085436, 69768478, 179982688, 464564220, 1200563864, 3104192722, 8034256412, 20803994184, 53915334890, 139785953076, 362681515714, 941361260956, 2444866458524, 6351963691964

Offset: 0

David W. Wilson

Siqi Wang, Table of n, a(n) for n = 0..40
A. J. Guttmann and G. M. Torrie, Critical behavior at an edge for the SAW and Ising model, J. Phys. A 17 (1984), 3539-3552. See series coefficients c_{n}^{2} for square lattice with wedge angle Pi/2.

Cf. A034010, A001411, A046170.

a(n) = 2 * A046170(n) for n >= 1. - Siqi Wang, Jul 15 2022

a(25)-a(32) from Bert Dobbelaere, Jan 05 2019

A064298 Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 4, 4, 1, 1, 8, 12, 8, 1, 1, 16, 38, 38, 16, 1, 1, 32, 125, 184, 125, 32, 1, 1, 64, 414, 976, 976, 414, 64, 1, 1, 128, 1369, 5382, 8512, 5382, 1369, 128, 1, 1, 256, 4522, 29739, 79384, 79384, 29739, 4522, 256, 1, 1, 512, 14934, 163496, 752061, 1262816, 752061, 163496, 14934, 512, 1

Offset: 1

Henry Bottomley, Sep 05 2001

			The start of the sequence as table:
* 1  1    1     1      1        1         1 ...
* 1  2    4     8     16       32        64 ...
* 1  4   12    38    125      414      1369 ...
* 1  8   38   184    976     5382     29739 ...
* 1 16  125   976   8512    79384    752061 ...
* 1 32  414  5382  79384  1262816  20562673 ...
* 1 64 1369 29739 752061 20562673 575780564 ...

S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 331-339.

Andrew Howroyd, Table of n, a(n) for n = 1..378
Steven R. Finch, Self-Avoiding Walks of a Rook on a Chessboard [From Steven Finch, Apr 20 2019]
Steven R. Finch, Self-Avoiding Walks of a Rook [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]
Steven R. Finch, Table of Non-Overlapping Rook Paths [From Steven Finch, Apr 20 2019; mentioned in Finch's "Gammel" link above]

A064297 together with its transpose.
Rows and columns include A000012, A000079, A006192, A007786, A007787, A145403, A333812.
Main diagonal is A007764.
Cf. A271465.

# Using graphillion
from graphillion import GraphSet
import graphillion.tutorial as tl
def A064298(n, k):
    if n == 1 or k == 1: return 1
    universe = tl.grid(n - 1, k - 1)
    GraphSet.set_universe(universe)
    start, goal = 1, k * n
    paths = GraphSet.paths(start, goal)
    return paths.len()
print([A064298(j + 1, i - j + 1) for i in range(11) for j in range(i + 1)])  # Seiichi Manyama, Apr 06 2020

A077482 Number of self-avoiding walks on square lattice trapped after n steps.

Original entry on oeis.org

1, 2, 11, 25, 95, 228, 752, 1860, 5741, 14477, 42939, 109758, 317147, 818229, 2322512, 6030293, 16900541, 44079555, 122379267, 320227677, 882687730, 2315257359, 6346076015, 16675422679, 45502168379, 119728011251, 325510252108, 857400725204

Offset: 7

Hugo Pfoertner, Nov 07 2002

Only 1/8 of all possible walks is counted by selecting the first step in +x direction and requiring the first step changing y to be positive.

			a(7) = 1 because there is only 1 self-trapping walk with 7 steps: (0,0)(1,0)(1,1)(1,2)(0,2)(-1,2)(-1,1)(0,1); a(8) = 2 because there are 2 self-trapping walks with 8 steps: (0,0)(1,0)(2,0)(2,1)(2,2)(1,2)(0,2)(0,1)(1,1) and (0,0)(1,0)(1,1)(2,1)(3,1)(3,0)(3,-1)(2,-1)(2,0).

See references given for A001411.

Hugo Pfoertner, Results for the 2D Self-Trapping Random Walk
Eric Weisstein's World of Mathematics, Self-Avoiding Walk.

Cf. A001411, A046661, A174517, A322831.

Fortran
```
c See Hugo Pfoertner link.
```

a(26)-a(28) from Alois P. Heinz, Jun 16 2011

a(29)-a(34) from Bert Dobbelaere, Jan 03 2019

A163529 The Y-coordinate of the n-th point in the Peano curve A163334.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 2, 2, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 5, 5, 4, 4, 4, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8, 7, 7, 7, 6, 6, 6, 6, 6, 6, 7, 7, 7, 8, 8, 8, 8, 8, 8

Offset: 0

Antti Karttunen, Aug 01 2009

There is a 2-state automaton that accepts exactly those pairs (n,a(n)) where n is represented in base 9 and a(n) in base 3; see accompanying file a163529.pdf. - Jeffrey Shallit, Aug 10 2023

Antti Karttunen, Table of n, a(n) for n = 0..59048
Jeffrey Shallit, Automaton for A163529
Index entries for sequences related to coordinates of 2D curves

Cf. A002262, A163335, A025581, A163337, A163326, A163332.

Cf. A059252, A059253, A163528, A163530, A163531, A163533.

a(n) = A002262(A163335(n)) = A025581(A163337(n)) = A163326(A163332(n)).

Name corrected by Kevin Ryde, Aug 28 2020

A335570 Number A(n,k) of n-step k-dimensional nonnegative lattice walks starting at the origin and using steps that increment all components or decrement one component by 1; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 3, 3, 1, 1, 1, 4, 7, 6, 1, 1, 1, 5, 13, 17, 10, 1, 1, 1, 6, 21, 40, 47, 20, 1, 1, 1, 7, 31, 81, 136, 125, 35, 1, 1, 1, 8, 43, 146, 325, 496, 333, 70, 1, 1, 1, 9, 57, 241, 686, 1433, 1753, 939, 126, 1, 1, 1, 10, 73, 372, 1315, 3476, 6473, 6256, 2597, 252, 1

Offset: 0

Alois P. Heinz, Jan 26 2021

			A(2,2) = 3: [(0,0),(1,1),(2,2)], [(0,0),(1,1),(0,1)], [(0,0),(1,1),(1,0)].
Square array A(n,k) begins:
  1,  1,   1,    1,    1,     1,     1,      1, ...
  1,  1,   1,    1,    1,     1,     1,      1, ...
  1,  2,   3,    4,    5,     6,     7,      8, ...
  1,  3,   7,   13,   21,    31,    43,     57, ...
  1,  6,  17,   40,   81,   146,   241,    372, ...
  1, 10,  47,  136,  325,   686,  1315,   2332, ...
  1, 20, 125,  496, 1433,  3476,  7525,  14960, ...
  1, 35, 333, 1753, 6473, 18711, 46165, 102173, ...
  ...

Alois P. Heinz, Antidiagonals n = 0..60, flattened

Columns k=0-10 give: A000012, A001405, A151265, A149424, A346225, A346226, A346227, A346228, A346229, A346230, A346231.
Rows n=0+1,2-3 give: A000012, A000027(k+1), A002061(k+1).
Main diagonal gives A335588.
Cf. A340591.

b:= proc(n, l) option remember; `if`(n=0, 1, b(n-1, map(x-> x+1, l))+add(
     `if`(l[i]>0, b(n-1, sort(subsop(i=l[i]-1, l))), 0), i=1..nops(l)))
    end:
A:= (n, k)-> b(n, [0$k]):
seq(seq(A(n, d-n), n=0..d), d=0..12);

b[n_, l_] := b[n, l] = If[n == 0, 1, b[n - 1, l + 1] + Sum[If[l[[i]] > 0, b[n - 1, Sort[ReplacePart[l, i -> l[[i]] - 1]]], 0], {i, 1, Length[l]}]];
A[n_, k_] := b[n, Table[0, {k}]];
Table[Table[A[n, d - n], {n, 0, d}], {d, 0, 12}] // Flatten (* Jean-François Alcover, Jan 29 2021, after Alois P. Heinz *)

A(n,k) == 1 (mod k) for k >= 2.

cp's OEIS Frontend

A151541 Number of 2-sided triangular strip polyedges with n cells.

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A152746 Six times hexagonal numbers: 6*n*(2*n-1).

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A276852 Number of positive walks with n steps {-3,-2,-1,1,2,3} starting at the origin, ending at altitude 1, and staying strictly above the x-axis.

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Mathematica

A001184 Number of simple Hamiltonian paths connecting opposite corners of a 2n+1 X 2n+1 grid.

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A010575 Number of n-step self-avoiding walks on 4-d cubic lattice.

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Fortran

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A038373 Number of n-step self-avoiding paths on quadrant grid starting at quadrant origin.

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A064298 Square array read by antidiagonals of self-avoiding rook paths joining opposite corners of n X k board.

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Python

A077482 Number of self-avoiding walks on square lattice trapped after n steps.

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Fortran

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A163529 The Y-coordinate of the n-th point in the Peano curve A163334.

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A152746 Six times hexagonal numbers: 6n(2*n-1).